Felhasználod az addiciós tételeket:
`cos(x+y)=cos(x)cos(y)-sin(x)sin(y)`
`sin(x+y)=sin(x)cos(y)+sin(y)cos(x)`
`cos(2*alpha)=cos^2(alpha)-sin^2(alpha)`
`sin(2*alpha)=2*sin(alpha)cos(alpha)`

Átalakítás közben ezt is használhatod:
`sin^2(x)+cos^2(x)=1`

`cos(3*alpha)=cos(alpha+2*alpha)=cos(alpha)*(cos^2(alpha)-sin^2(alpha))-sin(alpha)*2*sin(alpha)cos(alpha)=-3*cos(alpha)+4*cos^3(alpha)`.
`sin(3*alpha)=sin(alpha+2*alpha)=sin(alpha)(cos^2(alpha)-sin^2(alpha))+2*sin(alpha)cos(alpha)*cos(alpha)=3*sin(alpha)-4*sin^3(alpha)`.
`cos(4*alpha)=cos(2*alpha+2*alpha)=cos(2*alpha)*cos(2*alpha)-sin(2*alpha)sin(2*alpha)=(cos^2(alpha)-sin^2(alpha))^2-(2*sin(alpha)*cos(alpha))^2=`
`=8*cos^4(alpha)-8*cos^2(alpha)+1`.
`sin(4*alpha)=sin(2*2*alpha)=2*(sin(2*alpha)*(cos(2*alpha))=2*(2*sin(alpha)cos(alpha))*(cos^2(alpha)-sin^2(alpha))=`
`=4*cos^3(alpha)*sin(alpha)-4*cos(alpha)*sin^3(alpha)`.
`cos(5*alpha)=cos(alpha+4*alpha)=cos(alpha)*cos(4*alpha)-sin(alpha)*sin(4*alpha)=`
`=cos(alpha)*(8*cos^4(alpha)-8*cos^2(alpha)+1)-sin(alpha)*(4*cos^3(alpha)*sin(alpha)-4*cos(alpha)*sin^3(alpha))=`
`=8*cos^5(alpha)-8*cos^3(alpha)+cos(alpha)-4*cos^3(alpha)*sin^2(alpha)+4*cos(alpha)*sin^4(alpha)`.
`sin(5*alpha)=sin(alpha+4*alpha)=sin(alpha)*(8*cos^4(alpha)-8*cos^2(alpha)+1)+cos(alpha)*(4*cos^3(alpha)*sin(alpha)-4*cos(alpha)*sin^3(alpha))`.